Temperature Compensation in Wave-Based Damage Detection Systems

ABSTRACT

A method performed by a processing device, the method comprising: obtaining first waveform data indicative of traversal of a first signal through a structure at a first time; applying a scale transform to the first waveform data and the second waveform data; computing, by the processing device and based on applying the scale transform, a scale-cross correlation function that promotes identification of scaling behavior between the first waveform data and the second waveform data; performing one or more of: computing, by the processing device and based on the scale-cross correlation function, a scale factor for the first waveform data and the second waveform data; and computing, by the processing device and based on the scale-cross correlation function, a scale invariant correlation coefficient between the first waveform data and the second waveform data.

CLAIM OF PRIORITY

This application claims priority under 35 U.S.C. §119(e) to provisionalU.S. Patent Application 61/741,434, filed on Jul. 19, 2012, the entirecontents of which is hereby incorporated by reference.

BACKGROUND

Structural health monitoring systems (SHM) and other damage detectionsystems use guided wave inspection methods for testing large structures.Guided ultrasonic waves are popular because they travel through thethickness of the structure over long distances with little attenuation.This allows sensors to interrogate large areas all at once. However,guided waves are inherently multi-modal and dispersive in theirpropagation. Furthermore, structures' boundaries generate reflectionsand exchange energy between wave modes. These effects make theinterpretation of measured data difficult and necessitate the use ofbaseline measurements with no damage present.

Under ideal conditions, SHM techniques easily detect damage as a changein the propagation medium by performing a subtraction or time domaincorrelation with a baseline signal. The baseline reduces complexity byremoving effects from static sources, such as reflecting boundaries.Unfortunately, when the environmental conditions change, these methodsare unable to distinguish damage from benign effects.

Temperature Compensation

One of the most prominent environmental effects to distort signals istemperature fluctuation. Temperature has been shown to modify thevelocity of the guided wave modes, which stretches or scales themeasured time domain signal. Given an ultrasonic signal measurementx(t), a change in temperature T{•} can be modeled as a uniform scale

T{x(t)}=x(αt),  (1)

where α is an unknown scaling factor. This effect can be attributed totemperature's influence on the Young's modulus of the material.

However, this model is an approximation. In the presence of multi-modaland dispersive propagation, temperature's effects do not perfectlydilate or compress the signal. Even so, the effectiveness of thisapproximate model has been experimentally demonstrated in references.This implies that, by computing a scale-invariant measure of similaritybetween the baseline signal and measured data, one can compensate fortemperature's effects on the ultrasonic signals. Two prior arttechniques have been proposed for ultrasonic temperature compensation bycomputing a scale-invariant statistic: (a) optimal signal stretch, and(b) local peak coherence.

Optimal Signal Stretch

Optimal signal stretch (OSS) is prior art technique for ultrasonictemperature compensation that uses an exhaustive search optimizationstrategy for finding the scale factor α which minimizes the mean squarederror between baseline and measured data. OSS employs a finite libraryof K stretched baselines {s(α₁ t), s(α₂ t), . . . , s(α_(K) t)} andcomputes the mean squared error between the observed signal and eachelement in the library. The scale factor which minimizes the meansquared error is then declared to be the optimal choice over the givenset. As K→∞, OSS computes the optimal scale factor, over all possiblescale factors, that minimizes the mean squared error.

When the baseline and observed data are normalized to have zero meansand equal L² norms, minimizing the mean squared error is equivalent tomaximizing the sample Pearson product-moment correlation coefficient, orcorrelation coefficient for short. In the use of OSS, data is normalizedso that changes in the signal mean or amplitude, which generally do notcorrespond with damage, do not bias the results. With normalizing thedata, the optimal scale factor α chosen by OSS is defined by

$\begin{matrix}{{\alpha = {\arg {\max\limits_{k}{\int_{0}^{\infty}{\frac{\left( {{x(t)} - \mu_{x}} \right)\left( {{s\left( {\alpha_{k}t} \right)} - \mu_{s,k}} \right)}{\sigma_{x}\left( {\sigma_{s}/\sqrt{\alpha_{k}}} \right)}\ {t}}}}}},{1 < k < K},} & (2)\end{matrix}$

where β and σ represent the mean and L² norm of each signal,

$\begin{matrix}{{\mu_{x} = {\lim\limits_{T\rightarrow\infty}{\frac{1}{T}{\int_{0}^{T}{{x(t)}\ {t}}}}}}\mspace{11mu} \; {\sigma_{x} = {\sqrt{\int_{0}^{\infty}{{{{x(t)} - \mu_{x}}}^{2}\ {t}}}.}}} & (3)\end{matrix}$

The correlation coefficient between the optimally scaled baseline s(α t)and observed data x(t) can then be expressed as

$\begin{matrix}{{\varphi_{xs} = {\max\limits_{k}{\int_{0}^{\infty}{\frac{\left( {{x(t)} - \mu_{x}} \right)\left( {{s\left( {\alpha_{k}t} \right)} - \mu_{s,k}} \right)}{\sigma_{x}\left( {\sigma_{s}/\sqrt{\alpha_{k}}} \right)}\ {t}}}}},{1 < k < K},} & (4)\end{matrix}$

where value of φ_(xs) has a range −1≦φ_(xs)≦1. As K→∞, φ_(xs) becomesscale-invariant. So if x(t) and s(t) are scaled replicas of one another(i.e., x(t)=s(α t)), then the value of φ_(xs) is 1. Conversely, ifx(t)=−s(α t), then the value of φ_(xs) is −1. When there is no scalerelationship between the two signals, φ_(xs)=0. For finite values of Kand situations in which x(t)≈s(α t), φ_(xs) describes the degree oflinear correlation between x(t) and the optimally scaled baseline s(αt).

Since OSS is an optimal technique, assuming the uniform scaling modeldescribed in (1), system 110 uses it as a baseline for comparing thetechniques described herein and other techniques. Although effective,OSS is computationally inefficient. For a discrete signal of N samplesand a library of K stretched baselines, the OSS correlation coefficientcan be computed using a matrix-vector multiplication. Thecomputationally complexity of this operation, in big-O notational, isO(K N). Therefore, for sufficiently large signals and a sufficientlydense library, OSS becomes computationally impractical.

Local Peak Coherence

A second prior art approach to ultrasonic temperature compensation isreferred to as the local peak coherence (LPC) technique, which is usedto estimate the scale factor from local delay measurements. It has beenexperimentally shown that diffuse-like signals, cluttered by reflectionsand multi-path propagation, also exhibit almost perfect signalstretching behavior as temperature varies. Since the diffuse-likecondition ensures the measured signal to have a long and continuousduration, the scale factor between the two signals can be estimated froma series of local delay estimates. LPC assumes that, given an observedsignal x(t) and uniformly scaled replica x(t)=s(α t), the uniformscaling effect can be approximated as a delay in a small region aroundt=t₀,

x(t)=s(αt)≈s(t−(1+α)t ₀) as t→t ₀.  (5)

To estimate these delays, a small portion of each signal is windowedaround time t=t₀ and the delay between each windowed signal is computedas the argument that maximizes the standard cross-correlation(coherence) function,

$\begin{matrix}{{d_{w} = {\arg {\max\limits_{t}{\int_{0}^{\infty}{{x_{w}(\tau)}{s_{w}\left( {t + \tau} \right)}\ {\tau}}}}}},{1 < w < W},} & (6)\end{matrix}$

where d_(w) is the delay estimate for window w, s_(w)(t) and x_(w)(t)are windowed baseline and observed signals, and W is the number ofwindow positions. As the window moves across each signal, the delayvalue increases linearly according to the scale factor. The scaleparameter is then computed by performing a regression analysis on theestimated delays. Using the computed scale factor, the correlationcoefficient between the appropriately scaled baseline and measured datais then computed and used as the scale-invariant statistic.

As compared with OSS, LPC can be computed very quickly. The delayestimates may be computed using the fast Fourier transform (FFT)algorithm. Assuming, N_(w) is the number of discrete samples in thewindow and there are W different window positions, computing every delayrequires a computational complexity of O(W N_(w) log(N_(w))). Ingeneral, N_(w) will be small. So overall, the computational efficiencyis linearly dependent on W, and at a minimum, the scale factor can beestimated using W=2. However, this efficiency comes at the cost ofrobustness. The technique approximates uniform scaling as a series ofdelays, which, as shown in equation (5), is true asymptotically as t→t₀.In general, this may not always be a good approximation. Also, anyeffect that does not uniformly scale the signal may disrupt a portion ofa delay estimates and may significantly alter the scale factor estimate.For this reason, LPC is not robust and may be adversely affected by manyeffects, including the introduction of damage.

Therefore there is still a need for methods for compensation fortemperature variations in ultrasonic inspection systems that providehigh quality results and do so rapidly.

SUMMARY

In one aspect of the present disclosure, a method performed by one ormore processing devices comprises obtaining first waveform dataindicative of traversal of a first signal through a structure at a firsttime, wherein a first ambient temperature is associated with thestructure at the first time; obtaining second waveform data indicativeof traversal of a second signal through the structure at a second time,wherein a second ambient temperature is associated with the structure atthe second time, the first ambient temperature differs from the secondambient temperature, and a difference between the first ambienttemperature and the second ambient temperature causes a distortion ofthe second signal, relative to the first signal, as the second signaltraverses through the structure; applying a scale transform to the firstwaveform data and the second waveform data; computing, by the processingdevice and based on applying the scale transform, a scale-crosscorrelation function that promotes identification of scaling behaviorbetween the first waveform data and the second waveform data; performingone or more of: computing, by the processing device and based on thescale-cross correlation function, a scale factor for the first waveformdata and the second waveform data, with the scale factor beingindicative of an amount of variation between the first ambienttemperature and the second ambient temperature; and computing, by theprocessing device and based on the scale-cross correlation function, ascale invariant correlation coefficient between the first waveform dataand the second waveform data, with the scale invariant correlationcoefficient comprising a compensation statistic for compensating for thedistortion of the second signal, relative to the first signal, thatresults from variation in the first ambient temperature and the secondambient temperature.

Implementations of the disclosure can include one or more of thefollowing features. In some implementations, the method includesdetecting, based on the scale factor and the scale invariant correlationcoefficient, one or more areas of structural change in the structure. Inother implementations, the structural change comprises a degradation ofthe structure. In still other implementations, the scale invariantcorrelation coefficient is indicative of a measure of similarity betweenthe first waveform data and the second waveform data.

In some implementations, the scale factor comprises a value that is usedas a multiplier in scaling the first waveform data to correspond to thesecond waveform data. In still other implementations, the processingdevice is included in a wave-based damage detection system. In yet otherimplementations, the structure comprises a concrete structure. In stillother implementations, the scale-cross correlation function is computedin accordance with: x(t)⋄s_(α)(t)=s⁻¹{ s{x(t)}s{s(t)}}, wherein ⋄represents a scale cross-correlation operation; wherein the overbarrepresents a complex conjugation operation; wherein x(t) is the firstwaveform data; wherein s(t) is the second waveform data; wherein αincludes a scaling factor on s(t); wherein s{x(t)} is the scale domainof x(t); wherein s{s(t)} is the scale domain of s(t); and wherein S⁻¹{•}represents an inverse scale transform.

In yet other implementations, computing the scale-cross correlationfunction comprises: computing a product of a scale domain of the firstwaveform data and a scale domain of the second waveform data. In stillother implementations, computing the scale-cross correlation functioncomprises: resampling the first waveform data; resampling the secondwaveform data; applying an amplification factor to the resampled firstand second waveform data; and cross-correlating the amplified, resampledfirst and second waveform data. In still other implementations,computing the scale-invariant correlation coefficient comprises:determining an increased value of the scale cross-correlation functionin a stretch domain factor, relative to other values of the scalecross-correlation function in the stretch domain factor.

In some implementations, one or more of the first signal and the secondsignal is an ultrasonic wave signal. In still other implementations, thestructure comprises one or more of a pipe structure, a heatingstructure, one or more pipe structures in an oil refinery, one or morepipe structures in a chemical refinery, one or more pipe structures in agas refinery, one or more natural fuse pipelines, one or more oilpipelines, one or more heating pipe structures, one or more cooling pipestructures, one or more pipe structures in a nuclear power plant, one ormore pressure vessels, one or more concrete structures of a bridge, oneor more concrete structures of civil infrastructure, one or more portionof an airplace, one or more portions of an aerospace vehicle, one ormore portions of a submarine, and one or more metallic structures. Instill other implementations, the method includes identifying amaximization of the scale-cross correlation function in the stretchfactor domain; wherein computing the scale factor comprises: computing,by the processing device and based on the maximization of thescale-cross correlation function in the stretch factor domain, the scalefactor for the first waveform data and the second waveform data; andwherein computing the scale invariant correlation coefficient comprises:computing, by the processing device and based on the maximization of thescale-cross correlation function in the stretch factor domain, the scaleinvariant correlation coefficient between the first waveform data andthe second waveform data.

In still other implementations, the method includes identifying amaximization of the scale-cross correlation function in the scaletransform domain; wherein computing the scale factor comprises:computing, by the processing device and based on the maximization of thescale-cross correlation function in the scale transform domain, thescale factor for the first waveform data and the second waveform data;and wherein computing the scale invariant correlation coefficientcomprises: computing, by the processing device and based on themaximization of the scale-cross correlation function in the scaletransform domain, the scale invariant correlation coefficient betweenthe first waveform data and the second waveform data.

In still another aspect of the disclosure, one or more machine-readablemedia are configured to store instructions that are executable by aserver to perform operations including obtaining first waveform dataindicative of traversal of a first signal through a structure at a firsttime, wherein a first ambient temperature is associated with thestructure at the first time; obtaining second waveform data indicativeof traversal of a second signal through the structure at a second time,wherein a second ambient temperature is associated with the structure atthe second time, the first ambient temperature differs from the secondambient temperature, and a difference between the first ambienttemperature and the second ambient temperature causes a distortion ofthe second signal, relative to the first signal, as the second signaltraverses through the structure; applying a scale transform to the firstwaveform data and the second waveform data; computing, by the processingdevice and based on applying the scale transform, a scale-crosscorrelation function that promotes identification of scaling behaviorbetween the first waveform data and the second waveform data; performingone or more of: computing, by the processing device and based on thescale-cross correlation function, a scale factor for the first waveformdata and the second waveform data, with the scale factor beingindicative of an amount of variation between the first ambienttemperature and the second ambient temperature; and computing, by theprocessing device and based on the scale-cross correlation function, ascale invariant correlation coefficient between the first waveform dataand the second waveform data, with the scale invariant correlationcoefficient comprising a compensation statistic for compensating for thedistortion of the second signal, relative to the first signal, thatresults from variation in the first ambient temperature and the secondambient temperature. Implementations of this aspect of the presentdisclosure can include one or more of the foregoing features.

In still another aspect of the disclosure, an electronic system includesa server; and one or more machine-readable media configured to storeinstructions that are executable by the server to perform operationsincluding: obtaining first waveform data indicative of traversal of afirst signal through a structure at a first time, wherein a firstambient temperature is associated with the structure at the first time;obtaining second waveform data indicative of traversal of a secondsignal through the structure at a second time, wherein a second ambienttemperature is associated with the structure at the second time, thefirst ambient temperature differs from the second ambient temperature,and a difference between the first ambient temperature and the secondambient temperature causes a distortion of the second signal, relativeto the first signal, as the second signal traverses through thestructure; applying a scale transform to the first waveform data and thesecond waveform data; computing, by the processing device and based onapplying the scale transform, a scale-cross correlation function thatpromotes identification of scaling behavior between the first waveformdata and the second waveform data; performing one or more of: computing,by the processing device and based on the scale-cross correlationfunction, a scale factor for the first waveform data and the secondwaveform data, with the scale factor being indicative of an amount ofvariation between the first ambient temperature and the second ambienttemperature; and computing, by the processing device and based on thescale-cross correlation function, a scale invariant correlationcoefficient between the first waveform data and the second waveformdata, with the scale invariant correlation coefficient comprising acompensation statistic for compensating for the distortion of the secondsignal, relative to the first signal, that results from variation in thefirst ambient temperature and the second ambient temperature.Implementations of this aspect of the present disclosure can include oneor more of the foregoing features.

All or part of the foregoing can be implemented as a computer programproduct including instructions that are stored on one or morenon-transitory machine-readable storage media, and that are executableon one or more processors. All or part of the foregoing can also beimplemented as an apparatus, method, or electronic system that caninclude one or more processors and memory to store executableinstructions to implement the stated operations.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features,objects, and advantages will be apparent from the description anddrawings, and from the claims.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is block diagram of an example environment for generatingtemperature compensation indicators.

FIG. 2 shows a graph indicative of a relationship between a number ofsamples to resample a uniformly sampled signal and a first-orderapproximation.

FIGS. 3A-3C show graphs of sampling on a sine wave.

FIG. 4 shows a diagram that demonstrates actions in computing thediscrete-time exponential-time cross-correlation.

FIG. 5 shows curves demonstrating the computational growth rates of OSS(N² ln(N)) and the scale cross-correlation (N ln(N) log₂(N)).

FIG. 6 shows a graph of measured ambient temperatures during asimulation.

FIGS. 7A-7D show correlation coefficients for temperature compensationmethods.

FIGS. 8A-8B show diagrams of a wideband, modulated sinc waveform thatcaptures a frequency spectrum.

FIGS. 9A-9B show graphs indicative of a wideband response of a thinaluminum plate.

FIG. 10 shows correlation coefficients versus scale cross-correlationscale estimates.

FIGS. 11A, 11B are examples of graphs that illustrate the relationshipsbetween the standard and compensated correlation coefficients and thescale estimates.

FIG. 12 is a flowchart of a process for generating temperaturecompensation indicators.

FIG. 13 is a block diagram of components in an example environment forgenerating temperature compensation indicators.

DETAILED DESCRIPTION

A system disclosed herein computes a scale-invariant descriptor based onthe Mellin transform, to be used to improve wave-based damage detectionsystems by compensating for the variation in signals that result fromvariations in temperature. This compensation statistic is referred to asthe scale-invariant correlation coefficient, and the system measures itfrom the scale cross-correlation function between two signals.

Referring to FIG. 1, a block diagram is shown of an example environment100 for generating temperature compensation indicators, e.g., valuesindicative of an amount to compensate for the variation in signals thatresult from variations in temperature. There are various types oftemperature compensation indictors, including, e.g., a scale invariantcoefficient 115, a scale factor 117, and so forth.

Example environment 100 includes structure 102, e.g., a concretestructure, a pipe structure, a heating structure, a refinery, and anuclear power plant and so forth. Example environment 100 also includessystem 110 and data repository 114. In this example, data repository 114that stores information 109 indicative of various algorithms, including,e.g., algorithms that are represented by and/or in accordance with theequations described herein. In this example, system 110 uses information109 to compute various temperature compensation indicators.

In the example of FIG. 1, system 110 is configured to emit signals(e.g., ultrasonic wave signals) to structure 102 at various points intime, e.g., a first time (T1) and a second time (T2). Based on theemitted signals, system 110 records waveform data 104, 105 that isindicative of how signals 106, 107, respectively, travel throughstructure 102 at times T1, T2, respectively. In this example, system 110detects structural changes (e.g., damage) in structure 102 based onchanges in waveform data 104, 105.

In this example, the temperature of structure 102 (e.g., the ambient airsurrounding structure 102) fluctuates between time T1 and time T2. Thesefluctuations cause variations in the waveform data 104, 105. To accountfor these temperature based variations (which are independent from anddifferent from structural damage), system 110 uses the techniquesdescribed herein to compute temperature compensation indicators. Byapplication of the temperature compensation indicators to one or more ofwaveform data 104, 105, system 110 can identify the variations inwaveform data 104, 105 that are caused by temperature fluctuations andcompensate for them, e.g., thereby promote detection of variations inwaveform data 104, 105 that are caused by structural damage in structure102. The example environment 100 may include many thousands of clientdevices and systems.

In this example, one of the algorithms included in information 109 isbased on the Mellin transform, which is defined by the integral

M{x(t)}=X(p)=∫₀ ^(∞) x(t)t ^(p-1) dt.  (7)

In some examples, the special case for which p=−jc,

M{x(t)}=X(jc)=∫₀ ^(∞) x(t)t ^(−jc−1) dt.  (8)

is also often referred to as the Mellin transform. To avoid confusion,equation (8) is referred to as the Mellin transform and equation (7) asthe generalized Mellin transform. Another case of the generalized Mellintransform is known as the scale transform, which system 110 uses tocompute the scale-invariant correlation coefficient. Generally, a scaletransform includes a modification (e.g., a transform) between two spaces(e.g., a source space and a destination space) where the axis vectors ofthe source space are longer or shorter than the corresponding axisvectors in the destination space. This causes objects to stretch orshrink along the axes as they are transformed between the two spaces.

In signal processing and computer vision, the Mellin transform has beenutilized for its scale-invariant properties. A time-scale operation onthe signal x(t) is expressed as a phase change in the Mellin domain,

M{x(αt)}=X(jc)e ^(jc ln(α).)  (9)

The magnitude of the Mellin transform is invariant to changes in thetime-scale of a signal. This is analogous to the Fourier magnitudespectrum's shift invariance property.

The generalized Mellin transform is closely related to the Laplacetransform. Through algebraic manipulation, the expression for the Mellintransform can be rewritten as

M{x(t)}=X(p)=∫_(−∞) ^(∞) x(t)e ^(p ln(t)) d┌ ln(t)┘.  (10)

In equation (10), the generalized Mellin transform is equivalent to theLaplace transform, but integrated over logarithmic time. When p=−j c,the Mellin transform becomes equivalent to the Fourier transformintegrated over logarithmic time. This implies that, if an input signalx(t) is exponentially skewed by substituting t=e^(τ), the Mellintransform expression becomes

$\begin{matrix}\begin{matrix}{{M\left\{ {x(t)} \right\}} = {X\left( {j\; c} \right)}} \\{= {\int_{0}^{\infty}{{x\left( ^{\tau} \right)}^{{- j}\; c\; r}\ {\tau}}}} \\{{= {F\left\{ {x\left( ^{\tau} \right)} \right\}}},}\end{matrix} & (11)\end{matrix}$

where F {•} represents the Fourier transform. This property is the basisfor the fast Mellin transform algorithm.

In engineering, the Mellin transform's scale invariance property hasbeen used in a variety of applications. These applications include theclassification of ships from radar signals and the interpretation ofspeech waveforms. A variant known as the Fourier-Mellin transform isalso popular in several engineering fields. The Fourier-Mellin transformis computed from the Mellin transform of the Fourier magnitude spectrum.As a result, the magnitude of the Fourier-Mellin transform achievesinvariance to both shifting and scaling. In two dimensions, the Fourierspectrum is also polar transformed to achieve rotation invariance. TheFourier-Mellin transform has been used in applications for the analysisof speech and Doppler waveforms, as well as for pattern recognition andwatermark detection in images.

Sometimes, the magnitude of the Mellin transform is analyzed alone toachieve invariance. Unfortunately, this is less than ideal since itrequires the removal of phase information, which is necessary forgenerating complete descriptors of a signal. However, when comparing twosignals, the product of the Mellin transformed signals can be used tocompute the Mellin cross-correlation function. The maximum of the Mellincross-correlation is a measure of the scale-invariant similarity betweentwo signals and the location of the maximum can also be used to estimatethe scale factor between the signals, as described in further detailbelow. The Mellin cross-correlation is used in combination with theFourier-Mellin transform for image registration and pattern recognitionapplications.

Although the Mellin transform has found widespread use in manyapplications, there are several difficulties involved with implementingit. First, the Mellin transform of a signal is not guaranteed to exist.Second, in discrete-time, efficient computation of the discrete-timeMellin transform requires data to be exponentially sampled in time. Thisis often impractical to implement. Therefore, the Mellin transform isusually approximated using the fast Mellin transform algorithm, in whichuniformly sampled data is exponentially resampled. However, thisnon-uniform resampling procedure incurs its own Nyquist samplingrestrictions and complicates computation of the associatecross-correlation function.

In this example, the scale cross-correlation function andscale-invariant correlation coefficient, based on the scale-invariantproperties of Mellin transform, provide (and/or are used in) ultrasonictemperature compensation. Below, a case of the generalized Mellintransform known as the scale transform is described. The scale transformexists for all finite energy signals. System 110 derives thecontinuous-time scale cross-correlation and uses it to derive thescale-invariant correlation coefficient and associated scale factor.System 110 also commutates the scale transform and scalecross-correlation in discrete time, and derives the Mellin resamplingtheorem to address the numerical difficulties with the fast Mellintransform algorithm.

Compared with the temperature compensation methods of the prior art, thetechniques for temperature compensation described herein have increasedcomputational speed and robustness (e.g., relative to the computationalspeed and robustness of the prior art methods). In an example, thecontinuous-time scale-invariant correlation coefficient described hereinis mathematically equivalent to the OSS correlation coefficient as thesize of the library K→∞. This suggests the techniques described hereinto be optimal in terms of mean squared error. In an example, thescale-invariant correlation coefficient described herein can be computedvery efficiently due to its relationship with the Fourier transform. Inthis example, the scale-invariant correlation coefficient is aseffective as OSS, an optimal method, and more robust than LPC.

In an example, system 110 implements stretch-based, model-driventemperature compensation algorithms based on the scale transform. In thescale transform domain, system 110 manipulates the stretch factor ofsignals and compute quantities invariant to changes in that stretchfactor. System 110 implements three algorithms for temperaturecompensation based on these scale domain tools: the scale-invariantcorrelation (SIC) method, the iterative scale transform (IST) method,and the combined SIC/IST method.

The Scale Transform

In an example, system 110 uses the scale transform and scalecross-correlation function for computing the scale factor and thescale-invariant correlation coefficient between two signals. In general,the generalized Mellin transform of equation (7) and the Mellintransform of equation (8) of a signal may not converge. Underappropriate conditions, the generalized Mellin transform of a functionƒ(t) converges within some analytic strip a<Re{p}<b, where a and b arereal numbers. To promote convergence, system 110 uses the scaletransform. The scale transform is a special case of the generalizedMellin transform for which p=−j c+½,

S{x(t)}=X(c)=∫₀ ^(∞) x(t)t ^(−jc−1/2) dt,  (12)

where S{•} represents the scale transform. The transform is sometimesnormalized by 1/(2π)^(1/2) to preserve symmetry between it and itsinverse. This integral is also sometimes referred to as the Mellintransform or the Mellin transform of square-integrable functions.

The scale transform shares several useful properties with the Fouriertransform that are not present in the Mellin transform. For example, thekernel of the inverse scale transform is the complex conjugate of thekernel of the scale transform, as shown in the below equation (13):

$\begin{matrix}\begin{matrix}{{S^{- 1}\left\{ {X(c)} \right\}} = {x(t)}} \\{= {\frac{1}{2\; \pi}{\int_{- \infty}^{\infty}{{X(c)}t^{{j\; c} - {1\text{/}2}}\ {{c}.}}}}}\end{matrix} & (13)\end{matrix}$

In equation (13), S⁻¹{•} represents the inverse scale transform. Thescale transform also satisfies Parseval's theorem. This implies that theL² norm, or energy, of the signal is conserved between the time andscale domains, in accordance with the below equation (14):

∫₀ ^(∞) |x(t)|² dt=∫ _(−∞) ^(∞) |X(c)|² dc.  (14)

Therefore, given a finite energy signal, its scale transform and itsinverse always exists. Since system 110 is configured to processphysical signals that have finite energy, the scale transform may bedefined. Parseval's theorem then also implies that the magnitude of thescale transform exhibits invariance to energy-preserving scalingoperations, in accordance with the below equation (15):

S{√{square root over (α)}x(αt)}=X(c)e ^(jc ln(α).)  (15)

As with the Mellin transform, the scale transform is closely related toa logarithmic-time Fourier transform. By performing a change ofvariables, such that t=e^(τ) and dt=e^(τ) dτ, the scale transform can berepresented in accordance with the below equation (16):

$\begin{matrix}\begin{matrix}{{S\left\{ {x(t)} \right\}} = {X(c)}} \\{= {\int_{- \infty}^{\infty}{{x\left( ^{\tau} \right)}^{{({{{- j}\; c} + {1\text{/}2}})}\tau}\ {\tau}}}} \\{= {F\left\{ {^{{({1\text{/}2})}\tau}{x\left( ^{\tau} \right)}} \right\}}} \\{{F{\left\{ {\overset{\sim}{x}(\tau)} \right\}.}}}\end{matrix} & (16)\end{matrix}$

From this expression, system 110 may also represent the inverse scaletransform as indicated in the below equation (17)

s ⁻¹ {X(c);t}=e ^(−(1/2)ln(t)) F ⁻¹ {X(c);ln(t)}  (17)

In an example, system 110 may not compute the inverse scale transform,as performing the substitution of τ=ln(t) has several numericalcomplications. Instead, system 110 may extract the same informationusing the inverse Fourier transform.

In equation (16), {tilde over (x)}(τ) is an energy-preserved,exponentially skewed signal, as demonstrated by performing the samechange of variables, where t=e^(τ) and dt=e^(τ)dτ, to show

∫₀ ^(∞) |x(t)|² dt=∫ ₀ ^(∞) |e ^((1/2)τ) x(e ^(τ))|² dt.  (18)

Therefore, the scale transform is equivalent to the Fourier transform ofa signal that has been exponentially time-skewed with constant energy.When performing the inverse scale transform, system 110 can reverse thisprocess

{tilde over (x)}(τ)=F ⁻¹ {X(c)},  (19)

and then perform an energy-preserving logarithmic skew on {tilde over(x)}(τ), such that x(t)=e^((1/2)ln(t)){tilde over (x)}(ln(t)), to returnthe signal to its original domain.

Scale Cross-Correlation Function

As with the Mellin transform, system 110 defines a cross-correlationfunction between two signals and use it to compute the scale factor andscale-invariant descriptor of similarity between them.

Optimization in the Stretch Factor Domain α

The scale cross-correlation function of two signals x(t) and s(t) isdefined by the product of their scale domains, as indicated in the belowequation (20):

x(t)⋄s _(α)(t)=S ⁻¹{ S{x(t)} S{s(t)}},  (20)

where ⋄ represents the scale cross-correlation operation and the overbarrepresents a complex conjugation operation. In the time domain, thescale cross-correlation function is defined in accordance with the belowequation (21):

Φ_(xs)(α)=x(t)⋄s _(α)(t)=∫₀ ^(∞) x(t)s(αt)dt α>0.  (21)

In equation (21), if y(t)=x(t)⋄s(t) is the scale cross-correlation ofx(t) and s(t), then Φ_(xs)(t) is the inner product of every x(t) and s(αt) pair for values of a greater than zero.

In some examples, due to numerical complications, system 110 may notcompute the inverse scale transform. To avoid these complications,system 110 further manipulates equation (20) by substituting the inversescale transform relationship in equation (17) and expressing the scalecross correlation in the ln(a) domain as indicated in the below equation(22):

$\begin{matrix}\begin{matrix}{{\Phi_{xs}(\alpha)} = {^{{- {({1\text{/}2})}}\mspace{14mu} {\ln {\; \;}(\alpha)}}F^{- 1}\left\{ {{X*(c){S(c)}};{\ln \mspace{11mu} (\alpha)}} \right\}}} \\{= {\alpha^{{- 1}/2}{{\Psi_{xs}\left( {\ln \mspace{11mu} (\alpha)} \right)}.}}}\end{matrix} & (22)\end{matrix}$

The function ψ(ln(α)) is another representation for the scale-crosscorrelation in the ln(α) domain.

Computation of Scale Invariant Correlation (SIC) Method

For Ψ_(xs)(ln(α)), which may be computable by system 110, the naturallogarithm is monotonic and the maximum with respect to α is equivalentto the maximum with respect to ln(α). Therefore, the maximum correspondsto the same maximum in Φ_(xs)(ln(α)). Therefore, system 110 computes theoptimal stretch factor (e.g., scale factor) as indicated in the belowequation (23):

$\begin{matrix}{\hat{\alpha} = {{\exp\left( {\arg \; {\max\limits_{\ln \mspace{11mu} {(\alpha)}}{\Psi_{xs}\left( {\ln (\alpha)} \right)}}} \right)}.}} & (23)\end{matrix}$

System 110 defines a scale-invariant correlation coefficient definedgenerally by the below equation (24):

$\begin{matrix}\begin{matrix}{\varphi_{xs} = {\max\limits_{\alpha}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}{\int_{0}^{\infty}{x*(t){s\left( {\alpha \; t} \right)}\ {t}}}}}} \\{= {\max\limits_{\alpha}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}{\Phi_{xs}(\alpha)}}}} \\{{= {\max\limits_{\ln \mspace{11mu} {(\alpha)}}{\frac{1}{\sigma_{x}\sigma_{s}}{\Psi_{xs}\left( {\ln \mspace{11mu} (\alpha)} \right)}}}},}\end{matrix} & (24)\end{matrix}$

where φ_(xs) is normalized such that φ_(xs)=1 when x(t) is a stretchedreplica of s(t). The scale estimate {circumflex over (α)} (e.g., scalefactor) specifies how much a signal has scaled or stretch relative tothe baseline and the scale-invariant correlation coefficient φ_(xs)specifies how similar the two signals are, invariant to scaling orstretching.

Computation of Iterative Scale Transform (IST) Method—Maximization inthe Scale Transform Domain c

Instead of computing the estimated scale factor directly from thescale-cross correlation function, system 110 computes it through anoptimization in the scale transform domain c, e.g., by maximizing thescale-cross correlation function in the scale transforms domain c.

In an example, by applying Parseval's theorem of

${\int_{0}^{\infty}{x*(t){s(t)}\ {t}}} = {\frac{1}{2\; \pi}{\int_{- \infty}^{\infty}{X*(c){S(c)}\ {c}}}}$

and a time stretching property in accordance with S{√{square root over(α)}x(αt);c}=X(c)e^(jcln(α)) to the scale-cross correlation functionΦ_(xs)(α), system 110 expresses an optimal stretch estimate in the scaletransform domain (e.g., a scale factor) as indicated in the belowequation 25:

$\begin{matrix}\begin{matrix}{\hat{\alpha} = {\arg \; {\max\limits_{o}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}{\Phi_{xs}(\alpha)}}}}} \\{{\arg \; {\max\limits_{\alpha}{\frac{1}{2\; \pi \; \sigma_{x}\sigma_{s}}{\int_{- \infty}^{\infty}{X*(c){S(c)}^{j\; c\mspace{11mu} \ln \mspace{11mu} {(\alpha)}}\ {c}}}}}}}\end{matrix} & (25)\end{matrix}$

In this example, system 110 modifies the stretch factor α by alteringthe phase of the scale transform, e.g., by multiplying either X(c) orS(c) by a complex exponential.

The scale-invariant correlation coefficient may then be computed byusing from equation (25), in accordance with the below equation (26):

$\begin{matrix}{\varphi_{xs}\; = {\frac{1}{2\; \pi \; \sigma_{x}\sigma_{s}}{\int_{- \infty}^{\infty}{X*(c){S(c)}^{j\; c\mspace{14mu} \ln \mspace{11mu} {(\hat{\alpha})}}\ {{c}.}}}}} & (26)\end{matrix}$

These expressions provide another means of computing the scale factorestimate and scale-invariant correlation coefficient.

Performance of Scale Transform Methods

As discussed in further detail below, system 110 implements SICtechnique by computing the scale transform of each signal followed bycomputing the scale cross-correlation function, from which the scalefactor and scale-invariant correlation coefficient is extracted. System110 implements IST by computing the scale transform of each signal andthen uses a standard iterative optimization method to compute the scalefactor and scale-invariant correlation coefficient directly from thescale transforms. In an example, the SIC technique may be morecomputationally efficient, e.g., relative to computation efficiency ofthe IST technique. In still another example, the IST technique isassociated with increased resolution in terms of how accurate theresults may be, e.g., relative to the accuracy of the SIC technique.

Scale-Invariant Correlation (SIC) Method

The scale-invariant correlation (SIC) method maximizes Φ_(xs)(ln(α))directly in the log-stretch factor domain ln(a) as expressed in equation(23). In this example, by sampling x(t) and s(t) in the time domain andtruncating the signal to a length of N samples, the scale transformrepresentations, X(c) and S(c), are represented only by a finite numberof values. Using equation (23), system 110 computes Φ_(xs)(ln(α)) as theinverse Fourier transform of X*(c)S(c). In this example, system 110evaluates Φ_(xs)(ln(α)) for a finite, discrete set of stretch factors α.

The resolution of the set of stretch factors is defined by the samplinginterval of Φ_(xs)(ln(α)). Assuming a unitary sampling period, x(t) isdefined over the range 1≦t≦N and x(e^(T)) is defined for 0≦T≦ln(N). Inthis example, since x(e^(T)) is of length N ln(N), the interval betweeneach sample is 1/N. In this example, since Φ_(xs)(ln(α)) is related tox(e^(T)) and s(e^(T)) by a Fourier transform followed by an inverseFourier transform, it also has a sampling interval of 1/N.

This implies that the smallest measurable deviation from α=1 is

$\begin{matrix}{{\Delta \; \alpha} = {{\exp \left( {\pm \frac{1}{N}} \right)}.}} & (27)\end{matrix}$

For sufficiently large values of N, system 110 approximates Δα by afirst order Taylor series approximate to obtain

$\begin{matrix}{{\Delta \; \alpha} \approx {1 \pm {\frac{1}{N}.}}} & (28)\end{matrix}$

In this example, the resolution of SIC is approximately 1/N. Therefore,SIC is limited in resolution. However, system 110 improves thisresolution by combining SIC with an iterative optimization approach.

System 110 calculates the computational complexity of the SIC method. Tocompute SIC, system 119 exponentially resamples x(t) and s(t), computesFourier transforms of x(t) and s(t), compute an inverse Fouriertransform, and obtains a maximum of the result (and/or an increasedvalue). Maximizing Φ_(xs)(ln(α)) and exponentially resampling x(t) canboth be computed in linear time. In this example, x(e^(τ)) is of lengthN ln(N) and the computational complexity of computing its Fouriertransform, using the fast Fourier transform algorith, is O(N ln(N)log(Nln(N))), or O(N ln(N)log(N)) after simplifying. Since this is the mostcomputationally expensive operation in SIC, the computational complexityof SIC is also O(N ln(N)log(N)).

Iterative Scale Transform (IST) Method

The iterative scale transform (IST) method maximizes the scalecross-correlation function Φ_(xs)(α) by phase shifting X*(c) or S(c) inthe scale transform domain c as shown in equation (60). Solving thisoptimization problem iteratively in the scale transform domain c allowsIST to have increased precision. In an example, the scalecross-correlation is not (globally) convex, but is locally convex aroundmultiple maxima.

To compute the stretch factor estimate {circumflex over (α)} using IST,system 110 computes the scale transforms X*(c) and S(c). As with SIC,the complexity of these operation is O(N ln(N)log(N)). System 110selects an initial guess for α, multiply S(c) (or X*(c)) by e^(jcln(α)),and then computes the inner product between X*(c) and S(c)e^(jcln(α)).Each of these operations has a linear complexity. This process ofchoosing an α, applying a phase shift, and computing an inner product isthen repeated for different values of α by a vortex optimizationalgorithm until the inner product converges to a maximum value. Thecomplexity of most convex optimization algorithms, neglecting specialcases, is O(M²) where M is the number of parameters to optimize across.For this application, system 110 optimizes across one variable α, so M=1and the complexity is constant. Therefore, the complexity of theoptimization procedure is O(N ln(N)), for each iteration, where N ln(N)is the number of samples in the scale transform domain.

System 110 also improves the computational speed of IST by takingadvantage of the structure of the scale transform. The majority of theenergy in a signal is often located early in the scale transform domain.Therefore, system 110 truncates a large portion of the domain withlittle loss of information. As a result, the cost of the iterativealgorith becomes 0((ρN ln(N)), where ρ represents the percentage of thescale transform domain retained after truncation.

SIC/IST Combination

As previously discussed, IST is a very precise estimation strategy butonly if the result converges to the global maximum. In contrast, SICrequires no assumption of convexity but has a finite resolution. Bycombining these two methods, system 110 obtains highly precise estimatesand guarantees convergence to the global maximum. This is done by usingSIC to generate the initial stretch factor estimate {circumflex over(α)} for IST. In general, the SIC estimate will lie within the locallyconvexity region around the global maximum of the scalecross-correlation function Φ_(xs)(α).

In an example, if N is small enough such that SIC cannot adequatelyresolve the main lobe of the scale cross-correlation function, then theSIC estimate may not be accurate and IST may not be guaranteed toconverge to the globally optimal result. However, for sufficiently largevalues of N, this is not an issue.

Since IST computes the scale transform representations, X(c) and S(c),the only additional step required when combined with SIC is thecomputation of the inverse Fourier transform in the below equation (29)and maximization over Φ_(xs)(α) in equation (23).

$\begin{matrix}\begin{matrix}{\hat{\alpha} = {\arg \; {\max\limits_{\alpha}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}^{{- {({1\text{/}2})}}\mspace{14mu} \ln \mspace{11mu} {(\alpha)}}F^{- 1}\left\{ {{X*(c){S(c)}};{\ln \mspace{11mu} (\alpha)}} \right\}}}}} \\{= {\arg \; {\max\limits_{\alpha}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}F^{- 1}\left\{ {{X*(c){S(c)}};{\ln \mspace{11mu} (\alpha)}} \right\}}}}} \\{= {\arg \; {\max\limits_{\alpha}{\frac{\sqrt{\alpha}}{\sigma_{x}\sigma_{s}}{{\Psi_{x,s}\left( {\ln \mspace{11mu} (\alpha)} \right)}.}}}}}\end{matrix} & (29)\end{matrix}$

The computational complexity of these operations is O(N ln(N)log(N)),the same as initially computing X(c) and S(c). Therefore, theseoperations do not change the overall computational complexity of IST andthe complexity of SIC/IST is equivalent to the computational complexityof IST.

The properties of various temperature compensation techniques are shownin the below Table 1:

TABLE 1 Computational Iterative Methods Resolution Complexity ComplexityFinite Search OSS 1/N O(RN²) — Resolution SIC 1/N O(N ln(N)log(N)) —Methods Fine Search/ — O(RN²) O(N) Resolution Iterative OSS MethodsSIC/IST — O(N ln(N)log(N)) O(pN ln(N))

The Discrete-Time Scale Transform

In additional to processing continuous-time signals, system 110 is alsoconfigured to process discrete time signals that are sampled. As impliedfrom equations (11) and (18), a discrete-time Mellin transform ordiscrete-time scale transform (DTST) can be computed by exponentiallysampling a continuous-time signal and taking its discrete-time Fouriertransform (DTFT). Unfortunately, uniform sampling is usually performed.This presents several challenges in implementing the Mellin or scaletransforms.

If the samples are uniformly spaced, then the discrete-time Mellin orscale transforms must be approximated. One option is to replace thecontinuous-time signals with discrete-time signals and the integralswith summations. However, this provides a poor approximation of thetransform. In practice, this “direct'” approach has been shown to beless effective than other approximations and is slow, requiring O(N²)complexity. More often, the Mellin and scale transforms are computed byexponentially resampling the uniformly sampled signal and computing itsFourier transform with the FFT. This is known as the fast Mellintransform (FMT) algorithm.

Compared to the direct Mellin transform approximation, the fastapproximation is more accurate and computationally efficient. Theexponential resampling operation is implemented by interpolating unknownpoints on an exponential grid. According to the Nyquist-Shannon samplingtheorem, a band-limited signal may be perfectly reconstructed by sincinterpolation. However, performing sinc interpolation on an exponentialgrid is impractical and computationally expensive. Instead, cubic splineinterpolation is commonly used. Cubic spline interpolation provides aclose approximation to sinc interpolation and may be computed with O(M)complexity, where M is the number of points on the exponential grid. TheDTFT of the resampled signal then is computed using the FFT algorithm,which requires O(M log(M)) complexity.

Mellin Resampling Theorem

FIGS. 3A-3C illustrate examples of a sinusoid sampled on uniform and anexponential grid. Referring to FIG. 3A, graph 300 illustrates acontinuous-time sine wave. Referring to FIG. 3B, graph 302 illustrates auniformly sampled sine wave. Referring to FIG. 3C, graph 304 illustratesan exponentially sampled sine wave.

In generating an exponentially resampled signal, system 110 uses anexponential grid. To generate the exponential grid, system 110determines 1) where the first sample in the exponential grid is locatedin time; and 2) a number of samples to be included in the exponentialgrid. In this example, system 110 defines the first sample's locationand, from that result, derives an exact solution for the necessarynumber of samples.

In an example, signals may begin at time t=0, which is the scalingorigin. In another example, on an exponential grid, t=0 translates toτ=ln(0)=−∞. So as the first sample approaches t=0, the sampling intervalapproaches 0 and the number of necessary samples approaches infinity. Todefine the location of the first sample, system 110 enforces causalityin the exponential-time domain. A causal signal is nonzero for time τ≧0.In this example, causality is in exponential time, the signal x(t) willbe nonzero for values ln(t)≧0 or t≧1.

When exponential-time causality is assumed, the scale transform becomes

F{e ^((1/2)τ) x(e ^(τ))}=∫₀ ^(∞) e ^((1/2)τ) x(e ^(τ))e ^(−jc) dτ

S{x(t)}=∫₁ ^(∞) x(t)t ^(−jc−1/2) dt.  (30)

Exponential-time causality implies that the time signal x(t) begins attime t=1. In continuous time, this is not advantageous since it removesdata. However, for a uniformly sampled discrete-time signal, thisassumption is very practical. In discrete-time, t=nT, where T is thesampling interval. The sampling interval T scales the signal x[nT],which translates to a phase shift in the scale domain,

S{x[nT]}=T ^(−1/2) S{x[n]}e ^(jc ln(T)).  (31)

Therefore, when system 110 computes the DTST, system 110 determines (oraccesses information specifying that) the signal has a sampling rateof 1. Under a unitary sampling rate, t=1 occurs when n=1. So indiscrete-time, the causality condition allows us to exponentiallyresample starting from n=1 and neglect the asymptotic conditions aroundn=0, the scaling origin. This may require us to remove a single sample,assuming the first sample is located on the scaling origin. However,when analyzing the scaling behavior of a signal, the origin contains noinformation since it is not affected by scaling, x[α 0]=x[0], and thescale transform of values at the origin are not well defined. Forexample, the scale transform of a Dirac impulse δ(t) is a division byzero

$\begin{matrix}\begin{matrix}{{S\left\{ {\delta (t)} \right\}} = {\int_{0}^{\infty}{{\delta (t)}t^{{{- j}\; c} - {1\text{/}2}}\ {t}}}} \\{= {{\delta (t)}{(0)^{{{- j}\; c} - {1\text{/}2}}.}}}\end{matrix} & (32)\end{matrix}$

For these reasons, it is practical for us to assume the first sample isn=1 and the exponential-time domain is causal.

By enforcing causality, the exponentially resampled signal can becomputed with a finite number of samples. To properly resample thesignal, however, system 110 ensures that no information (or a reducedamount of information) is lost. This condition requires that the minimumsampling rate satisfies the Nyquist sampling criteria,

{tilde over (F)} _(s)≧2f _(max),  (33)

where {tilde over (F)}_(s) is the minimum sampling rate on theexponential grid and f_(max) is the largest non-zero frequency in thesignal. For example, if the uniform signal is critically sampled, wherethe sampling rate F_(s)=2f_(max), then the exponentially resampledsignal will require more samples than the uniform signal.

Mellin Resampling Theorem.

In this example, {tilde over (x)}[m] includes a causal, exponentiallyresampled signal generated from the uniformly sampled signal x[n], wherex[n] is sampled at the Nyquist rate F_(s), and exists for all integersin the domain 1≦n≦N. Assume {tilde over (x)}[m] to have a minimumsampling rate of {tilde over (F)}_(s) and include M samples. To promotemaintenance of the Nyquist criteria {tilde over (F)}_(s)≧F_(s) ismaintained, system 110 determines M in accordance with the belowequation (34):

$\begin{matrix}{M \geq {\frac{\ln \mspace{14mu} \left( {1/N} \right)}{\ln \mspace{14mu} \left( {1 - {1/N}} \right)} + 1.}} & (34)\end{matrix}$

In this example, without following the integer constraint, anexponentially resampled signal {tilde over (x)}[ñ] would exist acrossthe domain 0≦ñ≦ln(N). To satisfy the integer constraint, system 110determines ñ in accordance with the equation shown in the below equation(35):

$\begin{matrix}{{\overset{\sim}{n} = {{m\; \Lambda} = {m\left( \frac{\ln \mspace{11mu} (N)}{M - 1} \right)}}},{0 \leq m \leq {M - 1.}}} & (35)\end{matrix}$

The constant Λ represents the sampling interval for the exponentiallyresampled signal. Λ is referred to as the Mellin interval and itsinverse Λ⁻¹ as the Mellin rate. To then maintain the Nyquist criteria,the minimum sampling rate in {tilde over (x)}[m] must be greater than orequal to the Nyquist rate. This is the same as stating that maximumsampling interval, which lies between the last two samples asillustrated in FIG. 2, in {tilde over (x)}[m] must be less than or equalto one uniform sampling interval. Following that logic, system 110 makesa determination in accordance with the below equation (36):

$\begin{matrix}{{1 \geq {N - {\exp \left( {\left( {M - 2} \right)\frac{\ln \; (N)}{M - 1}} \right)}}}{1 \geq {N - N^{\frac{M - 2}{M - 1}}}}{N^{\frac{M - 3}{M - 1}} \geq {N - 1}}{{\left( \frac{M - 2}{M - 1} \right)\mspace{14mu} \ln \; (N)} \geq {\ln \; \left( {N - 1} \right)}}{{M\; \left( {{\ln \; (N)} - {\ln \; \left( {N - 1} \right)}} \right)} \geq {{2\mspace{14mu} \ln \; (N)} - {\ln \; \left( {N - 1} \right)}}}{M \geq \frac{{2\mspace{14mu} \ln \; (N)} - {\ln \left( {N - 1} \right)}}{{\ln (N)} - {\ln \; \left( {N - 1} \right)}}}{M \geq {\frac{\ln (N)}{\ln \left( \frac{N}{N - 1} \right)} + 1}}{M \geq {\frac{\ln \; \left( {1/N} \right)}{\ln \; \left( {1 - {1/N}} \right)} + 1.}}} & (36)\end{matrix}$

The Mellin Resampling Theorem provides an exact bound on the number ofsamples necessary to exponentially resample a signal, starting from n=1,while maintaining the Nyquist criteria. In practice, M is an integer, soa ceiling operation would be applied to the result. When a signal iscritically resampled in accordance with the below equation (37):

$\begin{matrix}{{M = {\frac{\ln \left( {1/N} \right)}{\ln \left( {1 - {1/N}} \right)} + 1}},} & (37)\end{matrix}$

the Mellin interval from equation (35) becomes

Λ=−ln(1−1/N).  (38)

In this example, M increases with N. In this example, ln(1−1/N) isreplaced with its first order Taylor series expansion −1/N. After makingthat substitution, the first-order approximation of equation (34)(34)becomes

$\begin{matrix}{{\frac{\ln \left( {1/N} \right)}{\ln \left( {1 - {1/N}} \right)} + 1} \approx {{N\mspace{14mu} \ln \; (N)} + 1.}} & (39)\end{matrix}$

The error function for this approximation is then defined by

$\begin{matrix}\begin{matrix}{{{g(N)} = {\frac{\ln \left( {1/N} \right)}{\ln \left( {1 - {1/N}} \right)} - {N\mspace{14mu} \ln \; (N)}}}\;} \\{= {{- {\ln (N)}}{\left( \frac{1 + {N\mspace{14mu} {\ln \left( {1 - {1/N}} \right)}}}{\ln \left( {1 - {1/N}} \right)} \right).}}}\end{matrix} & {(40)}\end{matrix}$

Assuming N>1, ln(1−1/N)<0 and N ln(1−1/N)<−1. From these inequalities,system 110 determines that the approximation error g(N)<0. In thisexample, under a first order approximation,

M>N ln(N)+1.  (41)

Therefore, M increases with N at a rate of approximately N ln(N).Referring to FIG. 2, diagram 200 shows the first-order approximationerror curve g(N). In general, the approximation is very close, deviatingby approximately 1.15 samples for every power of 10 samples N. This alsodemonstrates that the Mellin resampling theorem is consistent withprevious results, which demonstrated that, to maintain the Nyquistcriteria, M>N ln(N).

Discrete-Time Scale Transform

Using the Mellin resampling theorem in equation (34), the DTST isdefined as implemented by the fast Mellin transform algorithm. In thisexample, {tilde over (x)}[mΛ] includes the exponentially resampled andamplified signal generated from a uniformly sampled signal with aunitary sampling rate x[n],

{tilde over (x)}[mΛ]=e ^((1/2)mΛ) x[e ^(mΛ].)  (42)

The Mellin interval Λ is defined in equation (35). The DTST is thendefined as

$\begin{matrix}\begin{matrix}{{S\left\{ {x\lbrack n\rbrack} \right\}} = {X\left( ^{j\; c} \right)}} \\{= {\sum\limits_{m = o}^{\infty}\; {{x\left\lbrack {m\; \Lambda} \right\rbrack}^{{- j}\; c\; t}}}} \\{{= {F\left\{ {\overset{\sim}{x}\left\lbrack {m\; \Lambda} \right\rbrack} \right\}}},}\end{matrix} & (43)\end{matrix}$

where F now represents the DTFT of a discrete-time signal. The inversetransform can be computed by taking the inverse DTFT

{tilde over (x)}[mΛ]=F ⁻¹ {X(e ^(jc))}=∫_(−π) ^(π) X(e ^(jc))e ^(jct)de,  (44)

and logarithmically resampling and attenuating back to the originaldomain.

In another example, system 110 uses a perfect discrete-timeinterpolator. In this example, {tilde over (x)}[m] represents theenergy-preserving time-skewed replica of the original uniformly sampledsignal, such that

$\begin{matrix}{{{\sum\limits_{n = 0}^{\infty}\; {{x\lbrack n\rbrack}}^{2}} = \Lambda}{\sum\limits_{m = {- \infty}}^{\infty}\; {{{x\left\lbrack {\Lambda \; m} \right\rbrack}}^{2}.}}} & (45)\end{matrix}$

In this example, the actual energy in the exponentially resampled signalwill depend on the interpolation process used.

As in the continuous case, a scaling operation on a discrete-time signaltranslates to an energy-normalizing phase shift in the scale domain. Fora uniformly sampled signal, a scaling operation x[βn] becomes a shiftafter the exponential resampling process. Therefore, in the scaledomain, the scaling operation becomes

$\begin{matrix}\begin{matrix}{{S\left\{ {x\left\lbrack {\beta \; n} \right\rbrack} \right\}} = {F\left\{ {^{{({{- 1}\text{/}2})}\mspace{11mu} {\ln {(\beta)}}}{\overset{\sim}{x}\left\lbrack {{m\; \Lambda} + {\ln (\beta)}} \right\rbrack}} \right\}}} \\{= {\beta^{{- 1}\text{/}2}{X(c)}{^{{- j}\; c\mspace{14mu} \ln \mspace{11mu} {(\beta)}}.}}}\end{matrix} & (46)\end{matrix}$

Discrete-Time Scale Cross-Correlation Function

In this example, x[nT] and s[nT] include uniformly sampled signals withsampling interval T. The discrete-time Mellin cross-correlation betweenx[nT] and s[nT] is defined as

$\begin{matrix}\begin{matrix}{{{\Phi_{xs}\lbrack\alpha\rbrack} = {{x\lbrack{nT}\rbrack}{{\Diamond s}_{\alpha}\lbrack{nT}\rbrack}}},{\alpha > 0}} \\{{= {\sum\limits_{n = 0}^{\infty}\; {{x\lbrack{nT}\rbrack}{s\left\lbrack {\alpha \; {nT}} \right\rbrack}}}},{\alpha > 0}} \\{= {S^{- 1}{\left\{ {\overset{\_}{S\left\{ {x\lbrack{nT}\rbrack} \right\}}S\left\{ {s\lbrack{nT}\rbrack} \right\}} \right\}.}}}\end{matrix} & (47)\end{matrix}$

Since T scales the signal, it translates to a phase shift in the scaledomain so that

$\begin{matrix}\begin{matrix}{{\Phi_{xs}\lbrack\alpha\rbrack} = {S^{- 1}\left\{ {\overset{\_}{S\left\{ {x\lbrack{nT}\rbrack} \right\}}T^{{- 1}\text{/}2}^{{- j}\; c\mspace{11mu} \ln \mspace{11mu} {(T)}}S\left\{ {s\lbrack n\rbrack} \right\} T^{{- 1}\text{/}2}^{j\; c\mspace{11mu} \ln \mspace{11mu} {(T)}}} \right\}}} \\{= {T^{- 1}S^{- 1}{\left\{ {\overset{\_}{S\left\{ {x\lbrack{nT}\rbrack} \right\}}S\left\{ {s\lbrack n\rbrack} \right\}} \right\}.}}}\end{matrix} & (48)\end{matrix}$

In an example, discrete-time scale cross-correlation is defined from thepair of exponentially resampled and amplified signals, in accordancewith the below equations:

{tilde over (x)}[mΛ]=e ^((1/2)mΛ) x[e ^(mΛ])  (49)

{tilde over (s)}[mΛ]=e ^((1/2)mΛ) s[e ^(mΛ],)  (50)

where Λ is the Mellin interval defined in (38). From these signals,system 110 determines the exponentially-skewed scale cross-correlationfunction, in accordance with the below equation (51):

$\begin{matrix}\begin{matrix}{{\Phi_{xs}\left\lbrack {k\; \Lambda} \right\rbrack} = {T^{- 1}^{{- {({1\text{/}2})}}k\; \Lambda}{\sum\limits_{m = {- \infty}}^{\infty}\; {{\overset{\sim}{x}\left\lbrack {m\; \Lambda} \right\rbrack}{\overset{\sim}{s}\left\lbrack {{m\; \Lambda} + {k\; \Lambda}} \right\rbrack}}}}} \\{= {T^{- 1}^{{- {({1\text{/}2})}}k\; \Lambda}F^{- 1}\left\{ {\overset{\_}{F\left\{ {\overset{\sim}{x}\left\lbrack {m\; \Lambda} \right\rbrack} \right\}}F\left\{ {\overset{\sim}{s}\left\lbrack {m\; \Lambda} \right\rbrack} \right\}} \right\}}} \\{= {T^{- 1}^{{- {({1\text{/}2})}}k\; \Lambda}F^{- 1}\left\{ {\overset{\_}{S\left\{ {x\lbrack n\rbrack} \right\}}S\left\{ {s\lbrack n\rbrack} \right\}} \right\}}} \\{{= {^{{- {({1\text{/}2})}}k\; \Lambda}{\Psi_{xs}\left\lbrack {k\; \Lambda} \right\rbrack}}},}\end{matrix} & (51)\end{matrix}$

where kΛ=ln(α). System 110 defines Ψ_(xs)[kΛ] to be the exponential-timecross-correlation function in discrete time,

Ψ_(xs) [kΛ]=F ⁻¹ { S{x[nT]}S{s[nT]}}.  (52)

If {tilde over (x)}[mΛ] and {tilde over (s)}[mΛ] are nonzero within therange 0≦m≦M−1, then Φ_(xs)[kΛ] and Ψ_(xs)[kΛ] will be nonzero over−(M−1)≦k≦M−1. Direct computation of the scale cross-correlationΦ_(xs)[α] requires implementation of the inverse scale transform, andtherefore a logarithmic decimation of Φ_(xs)[kΛ]. However, Ψ_(xs)[kΛ] isnot a causal signal, an assumption made by the Mellin resamplingtheorem. In an example, system 110 computes the scale factor andscale-invariant correlation coefficient between two signals directlyfrom Ψ_(xs)[kΛ]. Referring to FIG. 4, process 400 illustrates actions inthe computation of the Ψ_(xs)[kΛ] from two windowed sinusoids, whichvary by a scale factor of 2. In this example, system 110 receives (402)sinusoids 403 a, 403 b. Sinusoid 403 a oscillates at 20 Hz. Sinusoid 403b oscillates at 40 Hz. System 110 exponentially resamples (404)sinusoids 403 a, 403 b, resulting in signals 405 a, 405 b, respectively.System 110 multiples (406) the resampled sinusoids by an exponentialamplification factor, resulting in signals 407 a, 407 b, respectively.System 110 also cross-correlates (408) the multiplied, resampledsinusoids with each other, resulting in signal 109.

Discrete-Time Scale Factor Computation

In this example, x[n] and s[n] include scaled replicas of each othersuch that x[n]=s[βn]. After exponentially resampling and amplifying thesignals, the scale becomes a shift

{tilde over (s)}[mΛ]=e ^((1/2)mΛ) s[e ^(mΛ])

{tilde over (x)}[mΛ]=(β)^(−1/2) {tilde over (s)}[mΛ+ln(β)]  (53)

Under these circumstances, system 110 applies the Cauchy-Schwarzinequality to the exponential-time cross-correlation function Ψ_(xs)[kΛ]in equation (52),

$\begin{matrix}\begin{matrix}{{\frac{1}{T^{2}\beta}{{\sum\limits_{m = {- \infty}}^{\infty}\; {{\overset{\sim}{s}\left\lbrack {{m\; \Lambda} + {\ln (\beta)}} \right\rbrack}{\overset{\sim}{s}\left\lbrack {\left( {m + k} \right)\Lambda} \right\rbrack}}}}^{2}} \leq} \\\begin{matrix}{\frac{1}{T^{2}\beta}{\sum\limits_{m = {- \infty}}^{\infty}{{{\overset{\sim}{s}\left\lbrack {{m\; \Lambda} + {\ln (\beta)}} \right\rbrack}}^{2} \cdot}}} \\{{{\sum\limits_{m = {- \infty}}^{\infty}\left. {{\overset{\sim}{s}\left\lbrack {m + k} \right)}\Lambda} \right)}}^{2} = {\frac{1}{T^{2}\beta}{{{\frac{1}{\Lambda}{\sum\limits_{m = {- \infty}}^{\infty}{{\overset{\sim}{s}\lbrack m)}}^{2}}}}^{2}.}}}\end{matrix}\end{matrix} & (54)\end{matrix}$

Assuming s[mΛ] is not a periodic function, the expressions in equation(54) will be equal when kΛ=ln(β). This implies that, for non-periodicfunctions, the scale cross-correlation has a unique maximum when

$\begin{matrix}{{\arg \; {\max\limits_{k}\mspace{14mu} {\Psi_{xs}\left\lbrack {k\; \Lambda} \right\rbrack}}} = {{\ln (\beta)}{\Lambda^{- 1}.}}} & (55)\end{matrix}$

Through algebraic manipulation, this concludes that the scale factor isequal to

$\begin{matrix}{\beta = {{\exp \left( {\Lambda \mspace{14mu} \arg \; {\max\limits_{k}\mspace{14mu} {\Psi_{xs}\left\lbrack {k\; \Lambda} \right\rbrack}}} \right)}.}} & (56)\end{matrix}$

The function Ψ_(xs)[kΛ] is generally not convex. Therefore, whenimplementing, care must be taken to compute the maximum.

Discrete-Time Scale-Invariant Correlation Coefficient

For simplicity, assume both s[nT] and x[nT] are zero-mean signals sothat their exponentially resampled versions {tilde over (s)}[mΛ] and{tilde over (x)}[mΛ] are also zero-mean. From the Cauchy-Schwarzinequality in equation (52), system 110 determines the maximum value ofthe exponential-time cross-correlation Ψ_(xs)[kΛ] in accordance with thebelow equation (55):

$\begin{matrix}\begin{matrix}{{\max\limits_{k}\mspace{11mu} {\Psi_{xs}\left( {k\; \Lambda} \right)}} = {\frac{1}{T\; \Lambda \sqrt{\beta}}{\sum\limits_{m = {- \infty}}^{\infty}\; {{\overset{\sim}{s}\lbrack m\rbrack}}^{2}}}} \\{= {\frac{1}{T}{\sqrt{\sum\limits_{m = {- \infty}}^{\infty}\; {{\overset{\sim}{s}\left\lbrack {m\; \Lambda} \right\rbrack}}^{2}} \cdot \sqrt{\sum\limits_{m = {- \infty}}^{\infty}\; {{\overset{\sim}{x}\left\lbrack {m\; \Lambda} \right\rbrack}}^{2}}}}} \\{= {\frac{1}{T}\sigma_{\overset{\sim}{s}}\sigma_{\overset{\sim}{x}}}}\end{matrix} & (57)\end{matrix}$

System 110 computes the discrete-time scale-invariant correlationcoefficient the in accordance with the below equation (58):

$\begin{matrix}{{\psi_{xs} = {\frac{T}{\sigma_{\overset{\sim}{s}}\sigma_{\overset{\sim}{x}}}\left( {\max\limits_{k}{\Psi_{xs}\left( {k\; \Lambda} \right)}} \right)}},} & (58)\end{matrix}$

As in continuous time, ψ_(xs) is 1 when x[nT]=s[βnT], ψ_(xs) is −1 whenx[nT]=−s[βnT], and there is no scale relationship between the twosignals when ψ_(xs)=0. System 110 uses the values between the extrema todetermine the degree scale-invariant linear correlation between the twosignals.

Simulation Setup and Methodology

In an example, system 110 is configured to compare the scalecross-correlation, LPC, and OSS compensation techniques using ultrasonicdata from a thin plate under variable temperature conditions. To produceguided waves, system 110 uses a pair of synchronized lead zirconatetitanate (PZT) piezoelectric transducers permanently bonded to thesurface of a 9.8 cm wide by 30.5 cm long by 0.1 cm aluminum plate.Signals were generated and measured at a sampling rate of 1 MHz using aNational Instruments PXI data acquisition equipment. For eachmeasurement, the transducers were driven using a wide-band impulse-likesinc excitation and its response recorded for 10 ms. Referring to FIGS.8A, 8B, diagrams 800, 802 show the excitation of a time domain signaland a power spectral density, respectively. Referring to FIGS. 9A, 9B,diagrams 900, 902 show the response (to excitation) of a time domainsignal and a power spectral density, respectively.

The experiment ran for 36.3 hours between 11:30 AM and 11:50 PM of thefollowing day. During this time, the aluminum plate was cooled andwarmed in a refrigerator while guided waves were generated andautomatically recorded by the transducers at 2 minute intervals. Theambient temperature within the refrigerator was also measuredautomatically at 1 minute intervals. Referring to FIG. 6, graph 600shows the measured ambient temperature over time. Starting at 11:30 AMon day one, the temperature was gradually reduced from room temperature,20.5° C., to 3.37° C. until 2:51 PM. Between 2:51 PM and 7:33 PM, thetemperature was then gradually increased back to room temperature, 20.4°C. After 7:33 PM, the ambient temperature decreased and then remainedrelatively constant, at around 3.89° C., between midnight and 10:36 AMthe next day. Between 10:36 AM and 5:56 PM, the temperature was againgradually increased to 20.94° C. and then gradually decreased to 4.17°C. until 11:50 PM, when the experiment ended.

At 6:18 PM on the first day, a cylindrical, steel, greased-coupled masswith a diameter of 3.8 cm and a length of 4.5 cm, was placed on top ofthe aluminum plate to simulate damage. At 4:04 PM on the second day, themass was removed from the plate. The mass was removed to confirm thatthe changes in the wave propagation were a result of the mass and not anartifact introduced by the placement procedure. The timing also allowedus to measure both an increase and decrease in temperature with andwithout the mass present. Using a mass in place of damage allows easyrepeatability of the experiment. While the mass may not perfectlysimulate damage, it changes the propagation environment in ways thattemperature will not. In this way, the mass acts similar to damage.

An impulsive signal was chosen in order to drive the PZT transducers toexcite as wide a frequency band as possible. Signal scaling occurs inboth the time and frequency domains. Therefore, limiting the observablefrequencies will reduce the accuracy of the uniform scaling model usedby the temperature compensation strategies.

After recording each guided wave record, the signals are high-passfiltered with a 3 kHz cutoff frequency and advanced 0.5 ms. Thehigh-pass filtering operation in done to remove systematic low frequencynoise in the system. This noise arises from various sources, such asmechanical vibrations and the pyroelectric effect. The time shiftoperation is necessary to remove group delay introduced by theexcitation signal. Failing to account for group delay will introduceerrors into the compensation strategies since the scaling origin willthen not be located at t=0.

Correlation Analysis

System 110 compares five different descriptors used for detecting themass: the correlation coefficient, the OSS correlation coefficient, twovariations on the LPC correlation coefficient, and the scale-invariantcorrelation coefficient. The correlation coefficient is a standardmeasure of similarity but is very sensitive to changes in scale. Incontrast, the OSS, LPC, and scale-invariant correlation coefficients arescale-invariant measures and more robust to changes in temperature.

For each LPC method, system 110 utilizes a different form of linearregression. The first uses a standard least-squares linear regression toestimate the scale factor. The second LPC method uses a robust linearregression method using MATLAB's robustfit function. This functionperforms an iteratively reweighted least-squares regression, which ismore robust to outlier data. In implementing each LPC method, system 110uses a local window size of 1000 samples with a hop size of 1 samplebetween each cross-correlation operation. A larger hop size wouldincrease the computational speed of the technique, but the hop size waskept to a minimum to maximize the accuracy of the linear regression. Incomputing the OSS correlation coefficient, system 110 accesses a libraryof 200 baselines over a scale range of 0.99 to 1.01, which is reasonablefor this application. The library was designed to have the sameresolution as the scale cross-correlation.

Referring to FIGS. 7A, 7B, graphs 700, 702 show the correlationcoefficients computed over the course of the experiment for twodifferent baseline signals, taken at 20.5° C. and 12.4° C. respectively.The first baseline is near the highest temperature measured and thesecond is approximately the median temperature. Two baselines illustratehow each method is affected by different ranges of temperature change.In both plots, the standard correlation coefficient, labeled “nocompensation,” is significantly affected by temperature. As thetemperature deviates from the baseline temperature, the correlationcoefficient decreases significantly. However, when the mass is added orremoved, the correlation coefficient shows minor changes.

The introduction of the mass causes significant changes for each of thecompensated correlation coefficients. However, the LPC correlationcoefficients are significantly affected by other effects as well. InFIG. 7A and FIG. 7B, the LPC correlation coefficients are almost equalto the OSS and scale-invariant correlation coefficients when thetemperature is close to the baseline temperature. However, when thetemperature deviates too far from the baseline, the LPC correlationcoefficients drop and vary erratically. When the mass is on the plate,this erratic behavior becomes even more prevalent.

The OSS and scale-invariant correlation coefficients of varysignificantly at the moment the mass is introduced. The two techniquesare almost equal to each other throughout the experiment. However, bothdescriptors, as well as LPC, do vary weakly with temperature. Thisindicates that it may be difficult to distinguish the mass from a sharpchange in temperature using the compensated correlation coefficientsalone. By utilizing both these scale-invariant descriptors as well as anestimate of the scale change, system 110 has an improved ability todetect the mass.

Temperature and Scale

Referring to FIGS. 7C and 7D, graphs 704, 706 show the scale factorestimates for each observed signal relative to two baseline signals.Graph 704 illustrates scale estimates with baseline at 20.5° C. Graph706 illustrates scale estimates with baseline at 13.1° C.

Each temperature compensation method is associated with an estimate.Comparison of FIGS. 7A-7D with FIG. 6 shows that the scale estimatesfrom OSS and the scale cross-correlation follow a similar trend as theambient temperature. The LPC technique estimate, however, is a poorpredictor of temperature when the current temperature is far from thebaseline temperature and when the mass is on the specimen. As with thecorrelation results, the OSS and scale cross-correlation produce almostidentical results, and the scale estimates are not largely affected bythe addition or removal of the mass.

Referring to FIG. 10, graph 1000 shows the relationship between scaleand temperature to be approximately linear, regardless of the state ofthe mass. The data, however, does exhibit a hysteresis, in which thescale value transitions slower than the temperature. This phenomenon isa result of the temperature sensor recording the ambient temperaturerather than the temperature of the specimen. The temperature of themetal specimen changes at a slower rate than the air surrounding it.Therefore the speed of the guided waves, and scale change in the signal,will change slower than the temperature as recorded by the sensor.

Mass Classification

As illustrated in the FIGS. 7A-7B, the scale-invariant correlationcoefficient is not perfectly invariant to temperature. However, in FIGS.7C-7D, the scale estimate is affected by temperature. Therefore, thecombination of these two descriptors can be used to provide a morecomplete description of the system than either one alone.

FIGS. 11A, 11B illustrate the relationships between the standard andcompensated correlation coefficients and the scale estimates, using abaseline signal at 13.1° C. Referring to FIG. 11A, graph 1100 shows therelationship between scale, estimated using the scale cross-correlation,and the standard correlation coefficient without temperaturecompensation. In this plot, there are little separation between the datataken with the mass present and the data taken without the mass.Referring to FIG. 11B, graph 1102 illustrates the relationship betweenthe scale-invariant correlation coefficient and its associated scaleestimate. In this plot, there is a clear separation between the datataken with and without the mass. Given a linear discriminator, the twoconditions can be classified with one hundred percent accuracy. Usingboth the scale-invariant correlation coefficient and scale estimateallows us to better differentiate model errors caused by temperature andmore significant deviations caused by damage or other effects. In thisexample, the two curves may eventually meet. This implies that, as thetemperature deviates further from the baseline, the mass will becomeincreasingly more difficult to detect.

System 110 implements a computationally efficient technique, the scalecross-correlation function, for analyzing the scaling behavior betweentwo signals. Through analytical and experimental comparison with theoptimal signal stretch method, the scale cross-correlation is an optimal(minimum mean squared error) temperature compensation method, for auniform scaling model for temperature. Given a baseline and observedsignal, system 110 computes the scale factor and the scale-invariantcorrelation coefficient between them. The computation is presented forcontinuous-time and discrete-time systems. In discrete time, system 110derives the Mellin resampling theorem, which states the minimum numberof samples necessary to exponentially resample a uniformly sampledsignal.

The estimate of the scale factor is a robust, linear predictor oftemperature in the system. The scale-invariant correlation coefficientwas shown to be approximately invariant to variations in temperature.Using both the scale factor and scale-invariant correlation coefficient,a mass, simulating damage, could be detected with a lineardiscriminator.

Since the scale cross-correlation requires no dense baseline library, itis a potentially very flexible tool for temperature compensation.Furthermore, its close relationship with the Fourier transformdemonstrates it to be a potentially powerful tool as well.

Referring to FIG. 12, system 110 executes process 1200 in computingtemperature compensation indicators, e.g., a scale factor, a scaleinvariant correlation coefficient, and so forth. In operation, system110 records (1202) a signal at a first point in time. The recordsignaled may include waveform data. The record signal is a signal thathas travelled through a structure. At a second point in time, system 110records (1204) another signal that has travelled through the structure.Between the first point in time and the second point in time, thestructure has experienced one or more structure changes (e.g., damage orother structural modifications).

Using the recorded signals, system 110 applies (1206) one or more timecompensation algorithms, including, e.g., an algorithm for computing thescale cross-correlation function (from the scale transform), analgorithm for computing the scale factor (e.g., from the scalecross-correlation function), an algorithm for computing thescale-invariant correlation coefficient (e.g., from the scalecross-correlation function), and so forth. System 110 computes (1208)time compensation indicators, e.g., using the time compensationalgorithms. System 110 computes various types of time compensationindicators, including, e.g., a scale factor, a scale-invariantcorrelation coefficient, and so forth. Using the time compensationindicators, system 110 detects (1210) damage in the structure. Forexample, system 110 applies the computed time compensation indicators tothe record signals to correct and/or identify variations in the signalsthat are caused by temperature fluctuations between times T1, T2. Afteridentifying the variations that are caused by temperature fluctuations,system 110 identifies other variations in the signals, with these othervariations being indicative of structural changes in the structure.

Referring to FIG. 13, components 1300 of an environment (e.g.,environment 100) for generating temperature compensation indicators.Client device 101 can be any sort of computing device capable of takinginput from a user and communicating over a network (not shown) withsystem 110 and/or with other client devices. In an example, clientdevice 101 is a device associated with structure 102 (FIG. 1) forrecording and/or tracking traversal of a signal through structure 102.For example, client device 101 can be a mobile device, a desktopcomputer, a laptop, a personal digital assistant (“PDA”), a server, anembedded computing system, a mobile device and so forth. Client device101 can include monitor 1108, which renders visual representations ofinterface 1106.

System 110 can be any of a variety of computing devices capable ofreceiving data, such as a server, a distributed computing system, adesktop computer, a laptop, a cell phone, a rack-mounted server, and soforth. System 110 may be a single server or a group of servers that areat a same location or at different locations.

System 110 can receive data from client device 101 via interfaces 1106,including, e.g., graphical user interfaces. Interfaces 1106 can be anytype of interface capable of receiving data over a network, such as anEthernet interface, a wireless networking interface, a fiber-opticnetworking interface, a modem, and so forth. System 110 also includes aprocessor 1002 and memory 1004. A bus system (not shown), including, forexample, a data bus and a motherboard, can be used to establish and tocontrol data communication between the components of server 110. In theexample of FIG. 13, memory 1004 includes data engine 1302 for executingthe techniques and operations described herein.

Processor 1002 may include one or more microprocessors. Generally,processor 1002 may include any appropriate processor and/or logic thatis capable of receiving and storing data, and of communicating over anetwork (not shown). Memory 1004 can include a hard drive and a randomaccess memory storage device, such as a dynamic random access memory,machine-readable media, or other types of non-transitorymachine-readable storage devices. Components 1300 also include datarepository 114, which is configured to store data collected throughsystem 110 and generated by system 110.

Embodiments can be implemented in digital electronic circuitry, or incomputer hardware, firmware, software, or in combinations thereof.Apparatus of the invention can be implemented in a computer programproduct tangibly embodied or stored in a machine-readable storage deviceand/or machine readable media for execution by a programmable processor;and method actions can be performed by a programmable processorexecuting a program of instructions to perform functions and operationsof the invention by operating on input data and generating output.

The techniques described herein can be implemented advantageously in oneor more computer programs that are executable on a programmable systemincluding at least one programmable processor coupled to receive dataand instructions from, and to transmit data and instructions to, a datastorage system, at least one input device, and at least one outputdevice. Each computer program can be implemented in a high-levelprocedural or object oriented programming language, or in assembly ormachine language if desired; and in any case, the language can be acompiled or interpreted language.

Suitable processors include, by way of example, both general and specialpurpose microprocessors. Generally, a processor will receiveinstructions and data from a read-only memory and/or a random accessmemory. Generally, a computer will include one or more mass storagedevices for storing data files; such devices include magnetic disks,such as internal hard disks and removable disks; magneto-optical disks;and optical disks. Computer readable storage media are storage devicessuitable for tangibly embodying computer program instructions and datainclude all forms of volatile memory such as RAM and non-volatilememory, including by way of example semiconductor memory devices, suchas EPROM, EEPROM, and flash memory devices; magnetic disks such asinternal hard disks and removable disks; magneto-optical disks; andCD-ROM disks. Any of the foregoing can be supplemented by, orincorporated in, ASICs (application-specific integrated circuits).

Other embodiments are within the scope and spirit of the descriptionclaims. In another example, due to the nature of software, functionsdescribed above can be implemented using software, hardware, firmware,hardwiring, or combinations of any of these. Features implementingfunctions may also be physically located at various positions, includingbeing distributed such that portions of functions are implemented atdifferent physical locations.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications can be made without departing fromthe spirit and scope of the processes and techniques described herein.In addition, the logic flows depicted in the figures do not require theparticular order shown, or sequential order, to achieve desirableresults. In addition, other steps can be provided, or steps can beeliminated, from the described flows, and other components can be addedto, or removed from, the described systems. Accordingly, otherembodiments are within the scope of the following claims.

What is claimed is:
 1. A method performed by a processing device, themethod comprising: obtaining first waveform data indicative of traversalof a first signal through a structure at a first time, wherein a firstambient temperature is associated with the structure at the first time;obtaining second waveform data indicative of traversal of a secondsignal through the structure at a second time, wherein a second ambienttemperature is associated with the structure at the second time, thefirst ambient temperature differs from the second ambient temperature,and a difference between the first ambient temperature and the secondambient temperature causes a distortion of the second signal, relativeto the first signal, as the second signal traverses through thestructure; applying a scale transform to the first waveform data and thesecond waveform data; computing, by the processing device and based onapplying the scale transform, a scale-cross correlation function thatpromotes identification of scaling behavior between the first waveformdata and the second waveform data; performing one or more of: computing,by the processing device and based on the scale-cross correlationfunction, a scale factor for the first waveform data and the secondwaveform data, with the scale factor being indicative of an amount ofvariation between the first ambient temperature and the second ambienttemperature; and computing, by the processing device and based on thescale-cross correlation function, a scale invariant correlationcoefficient between the first waveform data and the second waveformdata, with the scale invariant correlation coefficient comprising acompensation statistic for compensating for the distortion of the secondsignal, relative to the first signal, that results from variation in thefirst ambient temperature and the second ambient temperature.
 2. Thecomputer-implemented method of claim 1, further comprising: detecting,based on the scale factor and the scale invariant correlationcoefficient, one or more areas of structural change in the structure. 3.The computer-implemented method of claim 2, wherein the structuralchange comprises a degradation of the structure.
 4. Thecomputer-implemented method of claim 1, wherein the scale invariantcorrelation coefficient is indicative of a measure of similarity betweenthe first waveform data and the second waveform data.
 5. Thecomputer-implemented method of claim 1, wherein the scale factorcomprises a value that is used as a multiplier in scaling the firstwaveform data to correspond to the second waveform data.
 6. The methodof claim 1, wherein the processing device is included in a wave-baseddamage detection system.
 7. The method of claim 1, wherein the structurecomprises a concrete structure.
 8. The method of claim 1, wherein thescale-cross correlation function is computed in accordance with:x(t)⋄s _(α)(t)=S ⁻¹{ S{x(t)}{s(t)}}, wherein ⋄ represents a scalecross-correlation operation; wherein the overbar represents a complexconjugation operation; wherein x(t) is the first waveform data; whereins(t) is the second waveform data; wherein α includes a scaling factor ons(t); wherein S{x(t)} is the scale domain of x(t); wherein S{s(t)} isthe scale domain of s(t); and wherein S⁻¹{•} represents an inverse scaletransform.
 9. The method of claim 1, wherein computing the scale-crosscorrelation function comprises: computing a product of a scale domain ofthe first waveform data and a scale domain of the second waveform data.10. The method of claim 1, wherein computing the scale-cross correlationfunction comprises: resampling the first waveform data; resampling thesecond waveform data; applying an amplification factor to the resampledfirst and second waveform data; and cross-correlating the amplified,resampled first and second waveform data.
 11. The method of claim 1,wherein computing the scale-invariant correlation coefficient comprises:determining an increased value of the scale cross-correlation functionin a stretch domain factor, relative to other values of the scalecross-correlation function in the stretch domain factor.
 12. The methodof claim 1, wherein one or more of the first signal and the secondsignal is an ultrasonic wave signal.
 13. The method of claim 1, whereinthe structure comprises one or more of a pipe structure, a heatingstructure, one or more pipe structures in an oil refinery, one or morepipe structures in a chemical refinery, one or more pipe structures in agas refinery, one or more natural fuse pipelines, one or more oilpipelines, one or more heating pipe structures, one or more cooling pipestructures, one or more pipe structures in a nuclear power plant, one ormore pressure vessels, one or more concrete structures of a bridge, oneor more concrete structures of civil infrastructure, one or more portionof an airplace, one or more portions of an aerospace vehicle, one ormore portions of a submarine, and one or more metallic structures. 14.The method of claim 1, further comprising: identifying a maximization ofthe scale-cross correlation function in the stretch factor domain;wherein computing the scale factor comprises: computing, by theprocessing device and based on the maximization of the scale-crosscorrelation function in the stretch factor domain, the scale factor forthe first waveform data and the second waveform data; and whereincomputing the scale invariant correlation coefficient comprises:computing, by the processing device and based on the maximization of thescale-cross correlation function in the stretch factor domain, the scaleinvariant correlation coefficient between the first waveform data andthe second waveform data.
 15. The method of claim 1, further comprising:identifying a maximization of the scale-cross correlation function inthe scale transform domain; wherein computing the scale factorcomprises: computing, by the processing device and based on themaximization of the scale-cross correlation function in the scaletransform domain, the scale factor for the first waveform data and thesecond waveform data; and wherein computing the scale invariantcorrelation coefficient comprises: computing, by the processing deviceand based on the maximization of the scale-cross correlation function inthe scale transform domain, the scale invariant correlation coefficientbetween the first waveform data and the second waveform data.
 16. Anapparatus comprising: one or more processing devices; and one or moremachine-readable hardware storage devices storing instructions that areexecutable to cause the one or more processing devices to performoperations comprising: obtaining first waveform data indicative oftraversal of a first signal through a structure at a first time, whereina first ambient temperature is associated with the structure at thefirst time; obtaining second waveform data indicative of traversal of asecond signal through the structure at a second time, wherein a secondambient temperature is associated with the structure at the second time,the first ambient temperature differs from the second ambienttemperature, and a difference between the first ambient temperature andthe second ambient temperature causes a distortion of the second signal,relative to the first signal, as the second signal traverses through thestructure; applying a scale transform to the first waveform data and thesecond waveform data; computing, based on applying the scale transform,a scale-cross correlation function that promotes identification ofscaling behavior between the first waveform data and the second waveformdata; performing one or more of: computing, based on the scale-crosscorrelation function, a scale factor for the first waveform data and thesecond waveform data, with the scale factor being indicative of anamount of variation between the first ambient temperature and the secondambient temperature; and computing based on the scale-cross correlationfunction, a scale invariant correlation coefficient between the firstwaveform data and the second waveform data, with the scale invariantcorrelation coefficient comprising a compensation statistic forcompensating for the distortion of the second signal, relative to thefirst signal, that results from variation in the first ambienttemperature and the second ambient temperature.
 17. The apparatus ofclaim 16, wherein the operations further comprise: detecting, based onthe scale factor and the scale invariant correlation coefficient, one ormore areas of structural change in the structure.
 18. One or moremachine-readable hardware storage devices storing instructions that areexecutable to cause one or more processing devices to perform operationscomprising: obtaining first waveform data indicative of traversal of afirst signal through a structure at a first time, wherein a firstambient temperature is associated with the structure at the first time;obtaining second waveform data indicative of traversal of a secondsignal through the structure at a second time, wherein a second ambienttemperature is associated with the structure at the second time, thefirst ambient temperature differs from the second ambient temperature,and a difference between the first ambient temperature and the secondambient temperature causes a distortion of the second signal, relativeto the first signal, as the second signal traverses through thestructure; applying a scale transform to the first waveform data and thesecond waveform data; computing, based on applying the scale transform,a scale-cross correlation function that promotes identification ofscaling behavior between the first waveform data and the second waveformdata; performing one or more of: computing, based on the scale-crosscorrelation function, a scale factor for the first waveform data and thesecond waveform data, with the scale factor being indicative of anamount of variation between the first ambient temperature and the secondambient temperature; and computing based on the scale-cross correlationfunction, a scale invariant correlation coefficient between the firstwaveform data and the second waveform data, with the scale invariantcorrelation coefficient comprising a compensation statistic forcompensating for the distortion of the second signal, relative to thefirst signal, that results from variation in the first ambienttemperature and the second ambient temperature.
 19. The one or moremachine-readable hardware storage devices of claim 18, wherein theoperations further comprise: detecting, based on the scale factor andthe scale invariant correlation coefficient, one or more areas ofstructural change in the structure.